Abstract
For each positive integer m and each real finite dimensional Banach space X, we set \(\beta (X,m)\) to be the infimum of \(\delta \in (0,1]\) such that each set \(A\subset X\) having diameter 1 can be represented as the union of m subsets of A whose diameters are at most \(\delta \). Elementary properties of \(\beta (X,m)\), including its stability with respect to X in the sense of Banach-Mazur metric, are presented. Two methods for estimating \(\beta (X,m)\) are introduced. The first one estimates \(\beta (X,m)\) using the knowledge of \(\beta (Y,m)\), where Y is a Banach space sufficiently close to X. The second estimation uses the information about \(\beta _X(K,m)\), the infimum of \(\delta \in (0,1]\) such that \(K\subset X\) is the union of m subsets having diameters not greater than \(\delta \) times the diameter of K, for certain classes of convex bodies K in X. In particular, we show that \(\beta (l_p^3,8)\le 0.925\) holds for each \(p\in [1,+\infty ]\) by applying the first method, and we proved that \(\beta (X,8)<1\) whenever X is a three-dimensional Banach space satisfying \(\beta _X(B_X,8)<\frac{221}{328}\), where \(B_X\) is the unit ball of X, by applying the second method. These results and methods are closely related to the extension of Borsuk’s problem in finite dimensional Banach spaces and to C. Zong’s computer program for Borsuk’s conjecture.
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Acknowledgements
The authors are grateful to Professor Chuanming Zong for his supervision and discussion, and to Thomas Jenrich for his useful remark on the state-of-the-art of Borsuk’s conjecture.
Funding
This work is supported by the National Natural Science Foundation of China (grant numbers: 11921001 and 12071444), the National Key Research and Development Program of China (2018YFA0704701), and the Natural Science Foundation of Shanxi Province of China (201901D111141).
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This work is supported by the National Natural Science Foundation of China (Grant numbers: 11921001 and 12071444), the National Key Research and Development Program of China (2018YFA0704701), and the Natural Science Foundation of Shanxi Province of China (201901D111141)
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Lian, Y., Wu, S. Partition Bounded Sets Into Sets Having Smaller Diameters. Results Math 76, 116 (2021). https://doi.org/10.1007/s00025-021-01425-2
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DOI: https://doi.org/10.1007/s00025-021-01425-2
Keywords
- Banach-Mazur metric
- Borsuk’s Conjecture
- complete sets
- convex bodies
- finite-dimensional Banach spaces
- simplices